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Strange matrix problem

  1. Apr 8, 2010 #1
    Hi,

    1. The problem statement, all variables and given/known data

    I was given a linear equation system of the form Ax=b where x=(x, y, z, w),
    I reduced (A|b) to its canonical form which is this:

    2. Relevant equations

    1_140250080.gif

    3. The attempt at a solution

    At first I thought that this means that there is an infinite number of solutions with one freedom degree.
    But I cannot imagine in the world what the solution vector might look like.
    I am sorry if this seems like a stupid question but can anyone plz explain this to me?

    Thanks
     

    Attached Files:

  2. jcsd
  3. Apr 8, 2010 #2

    Mark44

    Staff: Mentor

    Just read them off your final augmented matrix: x = 0, y = 0, z = 0; w is arbitrary. The solution represents the line in four-dimensional space that coincides with the w-axis.
     
  4. Apr 8, 2010 #3
    Do you mean that if we assign w=t a general solution could be V=(0,0,0,t)?
     
  5. Apr 8, 2010 #4

    Mark44

    Staff: Mentor

    Yes.
     
  6. Apr 8, 2010 #5
    O.K Iunderstand the technical side of what you are saying Av=0, but the equation you get is 0w=0. How can this have any meaning?
     
  7. Apr 8, 2010 #6

    Mark44

    Staff: Mentor

    Of course the equation 0w = 0 has meaning. It's true for any value of w.

    The matrix equation you were working on was Ax = 0, so what you were finding was the nullspace of matrix A. The nullspace of A (or null(A)) maps all vectors of the form <0, 0, 0, t>T to the zero vector in R4.
     
  8. Apr 8, 2010 #7
    Got it now, thanks!
     
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